Hammond UW-L Journal of Undergraduate Research XIV (2011)
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He mathematically defined the first six harmonics as ratios of the fundamental frequency, 1/1, 2/1, 3/1, 4/1, 5/1, and
6/1, or the first six integer multiples of the original tone’s frequency [Forster]. Mersenne worked out tuning systems
(this idea will be discussed below) based on harmonics. Also attributed with defining harmonics is Jean-Philippe
Rameau (1683-1764), a French composer and music theorist. Rameau understood harmonics in relation to
consonances and dissonances (intervals that sound good or clash) and harmonies. His paper, Treatise on Harmony,
published in the 1720’s, was a theory of harmony based on the fact that he heard many harmonics sounding
simultaneously when each note was played. Whether these two men’s work was related or not, Rameau’s Treatise
on Harmony created a stir that initiated a revolution in music theory. Musicians began to notice other harmonics
sounding in addition to the played, fundamental tone, notably the 12
th
and 17
th
, which are the 12
th
and 17
th
steps in
the scale of a given note, respectively
[Sawyer].
In the 18
th
century, calculus became a tool, and was used in discussions on vibrating strings. Brook Taylor,
who discovered the Taylor Series, found a differential equation representing the vibrations of a string based on
initial conditions, and found a sine curve as a solution to this equation [Archibald].
Daniel Bernoulli (1700-1782) and Leonhard Euler (1707-1783), Swiss mathematicians, and Jean-Baptiste
D’Alembert (1717-1783), a French mathematician, physicist, philosopher, and music theorist, were all prominent in
the ensuing mathematical music debate. In 1751, Bernoulli’s memoir of 1741-1743 took Rameau’s findings into
account, and in 1752, D’Alembert published Elements of theoretical and practical music according to the principals
of Monsieur Rameau, clarified, developed, and simplified. D’Alembert was also led to a differential equation from
Taylor’s problem of the vibrating string,
=
where the origin of the coordinates is at the end of the string, the x-axis is the direction of the string, y is the
displacement at time t [Archibald]. This equation is basically the wave equation, which will be discussed later.
Euler questioned the generality of this equation; while D’Alembert assumed one equation to represent the
string, Euler said it could lie along any arbitrary curve initially, and therefore require multiple different expressions
to model the curve. The idea was that a simply plucked string at starting position represents two lines, which cannot
be represented in one equation.
Bernoulli disagreed. After following Rameau’s hint, he made an arbitrary mix of harmonics to get =
+ 22 + 33 + [Sawyer]. His theory was that this equation could represent
every possible vibration that could be made by a stretched string released from some position. Setting t = 0 should
give the initial position of the string. Bernoulli said his solution was general, and therefore should include the
solutions of Euler and D’Alembert. This led to the problem of expanding arbitrary functions with trigonometric
series. This idea was received with much skepticism, and no mathematician would admit its possibility until it was
thoroughly demonstrated by Fourier [Archibald].
This leads us to our celebrity, Jean Baptiste Fourier, Baron de Fourier (1768-1830), a French
mathematician. Fourier, the ninth child of a tailor, originally intended to become a priest, but decided to study
mathematics instead. He studied at the military school in Auxerre, and was a staff member at the École Normale,
and then the École Polytechnique in Paris, and through a recommendation to the Bishop of Auxerre, he was
educated by the Benvenistes, a wealthy, scholarly family. He succeeded Lagrange at the École Polytechnique and
travelled to Egypt in 1798 with Napoleon, who made him governor of Lower Egypt. He returned to France in 1801
and published his paper On the Propagation of Heat in Solid Bodies in 1807 [Marks]. His theory about the solution
to a heat wave equation, stating the wave equation could be solved with a sum of trigonometric functions, was
criticized by scientists for fifteen years [Jordan]. What he came up with, effectively, was the Fourier Series. In 1812,
the memoir of his results was crowned by the French Academy [Archibald].
: The shape of a simply plucked string based on Euler’s ideas.
http://www.marco-learningsystems.com/pages/sawyer/music.html
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